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Lattice associated to 4th Jacobi theta function?

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$begingroup$

For a lattice (specifically the dual lattice of a torus) there is associated a theta function

$ theta_{Gamma}(w)=sum_{gammainGamma}w^{||gamma||^2},text{ where $w=e^{-4pi^2t}$ and $tin(0,infty)$.} $

In particular the lattices $mathbb{Z}$ and $mathbb{Z}+1/2$ have theta functions equal to

$theta_{mathbb{Z}}(w)=sum_{kinmathbb{Z}}w^{k^2}=1+2w+2w^4+2w^9+dots$ ,

and $theta_{mathbb{Z}+1/2}(w)=sum_{kinmathbb{Z}}w^{(k+1/2)^2}=2w^{1/4}+2w^{9/4}+2w^{25/4}+dots,$ respectively.

These correspond respectively to the third and second Jacobi theta functions, $theta_3(w)$ and $theta_2(w)$.

My question is this: is there a lattice corresponding to the fourth Jacobi theta function
$theta_4(w)=sum_{kinmathbb{Z}}(-w)^{k^2}$?

special-functions integer-lattices theta-functions

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edited yesterday

J. M. is not a mathematician

61.3k5152290

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0

$begingroup$

For a lattice (specifically the dual lattice of a torus) there is associated a theta function

$ theta_{Gamma}(w)=sum_{gammainGamma}w^{||gamma||^2},text{ where $w=e^{-4pi^2t}$ and $tin(0,infty)$.} $

In particular the lattices $mathbb{Z}$ and $mathbb{Z}+1/2$ have theta functions equal to

$theta_{mathbb{Z}}(w)=sum_{kinmathbb{Z}}w^{k^2}=1+2w+2w^4+2w^9+dots$ ,

and $theta_{mathbb{Z}+1/2}(w)=sum_{kinmathbb{Z}}w^{(k+1/2)^2}=2w^{1/4}+2w^{9/4}+2w^{25/4}+dots,$ respectively.

These correspond respectively to the third and second Jacobi theta functions, $theta_3(w)$ and $theta_2(w)$.

My question is this: is there a lattice corresponding to the fourth Jacobi theta function
$theta_4(w)=sum_{kinmathbb{Z}}(-w)^{k^2}$?

special-functions integer-lattices theta-functions

share|cite|improve this question

edited yesterday

J. M. is not a mathematician

61.3k5152290

asked yesterday

plebmaticianplebmatician

359112

$endgroup$

add a comment | 

$begingroup$

For a lattice (specifically the dual lattice of a torus) there is associated a theta function

$ theta_{Gamma}(w)=sum_{gammainGamma}w^{||gamma||^2},text{ where $w=e^{-4pi^2t}$ and $tin(0,infty)$.} $

In particular the lattices $mathbb{Z}$ and $mathbb{Z}+1/2$ have theta functions equal to

$theta_{mathbb{Z}}(w)=sum_{kinmathbb{Z}}w^{k^2}=1+2w+2w^4+2w^9+dots$ ,

and $theta_{mathbb{Z}+1/2}(w)=sum_{kinmathbb{Z}}w^{(k+1/2)^2}=2w^{1/4}+2w^{9/4}+2w^{25/4}+dots,$ respectively.

These correspond respectively to the third and second Jacobi theta functions, $theta_3(w)$ and $theta_2(w)$.

My question is this: is there a lattice corresponding to the fourth Jacobi theta function
$theta_4(w)=sum_{kinmathbb{Z}}(-w)^{k^2}$?

special-functions integer-lattices theta-functions

share|cite|improve this question

edited yesterday

J. M. is not a mathematician

61.3k5152290

asked yesterday

plebmaticianplebmatician

359112

$endgroup$

share|cite|improve this question

edited yesterday

J. M. is not a mathematician

61.3k5152290

asked yesterday

plebmaticianplebmatician

359112

share|cite|improve this question

edited yesterday

J. M. is not a mathematician

61.3k5152290

asked yesterday

plebmaticianplebmatician

359112

edited yesterday

J. M. is not a mathematician

61.3k5152290

edited yesterday

J. M. is not a mathematician

61.3k5152290

asked yesterday

plebmaticianplebmatician

359112

asked yesterday

plebmaticianplebmatician

359112